Besse conjecture with positive isotropic curvature

TL;DR

This paper proves the Besse conjecture for compact manifolds with positive isotropic curvature, showing that solutions to the critical point equation must be Einstein and isometric to a sphere.

Contribution

It establishes the validity of the Besse conjecture under the condition of positive isotropic curvature, a significant geometric restriction.

Findings

Solutions are Einstein metrics when positive isotropic curvature is assumed.

The manifold is isometric to a standard sphere under the given conditions.

Confirms the conjecture for a broad class of curvature conditions.

Abstract

The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function $f$ on the manifold that satisfies the following $$ (1+f){\rm Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. $$ It has been conjectured that if $(g, f)$ is a solution of the critical point equation, then $g$ is Einstein and so $(M, g)$ is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.